Chaotic Behavior Research Articles

Chaotic Dynamics of the Fractional Order Predator-Prey Model Incorporating Gompertz Growth on Prey with Ivlev Functional Response

This paper examines dynamic behaviours of a two-species discrete fractional order predator-prey system with functional response form of Ivlev along with Gompertz growth of prey population. A discretization scheme is first applied to get Caputo fractional differential system for the prey-predator model. This study identifies certain conditions for the local asymptotic stability at the fixed points of the proposed prey-predator model. The existence and direction of the period-doubling bifurcation, Neimark-Sacker bifurcation, and Control Chaos are examined for the discrete-time domain. As the bifurcation parameter increases, the system displays chaotic behaviour. For various model parameters, bifurcation diagrams, phase portraits, and time graphs are obtained. Theoretical predictions and long-term chaotic behaviour are supported by numerical simulations across a wide variety of parameters. This article aims to offer an OGY and state feedback strategy that can stabilize chaotic orbits at a precarious equilibrium point.

Node and edge centrality based failures in multi-layer complex networks

Multi-layer complex networks (MLCN) appears in various domains, such as, transportation, supply chains, etc. Failures in MLCN can lead to major disruptions in systems. Several research have focussed on different kinds of failures, such as, cascades, their reasons and ways to avoid them. This paper considers failures in a specific type of MLCN where the lower layer provides services to the higher layer without cross layer interaction, typical of a computer network. A three layer MLCN is constructed with the same set of nodes where each layer has different characteristics, the bottom most layer is Erdos–Renyi (ER) random graph with shortest path hop count among the nodes as gaussian, the middle layer is ER graph with higher number of edges from the previous, and the top most layer is preferential attachment graph with even higher number of edges. Both edge and node failures are considered. Failures happen with decreasing order of centralities of edges and nodes in static batch mode and when the centralities change dynamically with progressive failures. Emergent pattern of three key parameters, namely, average shortest path length (ASPL), total shortest path count (TSPC) and total number of edges (TNE) for all the three layers after node or edge failures are studied. Extensive simulations show that all but one parameters show definite degrading patterns. Surprising, ASPL for the middle layer starts showing a chaotic behaviour beyond a certain point for all types of failures.

Numerical Exploration of a Network of Nonlocally Coupled Josephson Junction Spurred by Wien BridgeOscillators

This paper focuses on the numerical study of the collective dynamics of nonlocally coupled Josephson junctions (JJ) driven by Wien bridge oscillators (JJSWBOs). A single JJSWBO exhibits monostable and bistable chaotic behaviors, bistable period-3 oscillations, and a coexistence of regular and chaotic dynamics. To explore the network's collective dynamics, the local dynamics are set in a bistable regime, with initial conditions ensuring random distribution of units on these two types of behavior. Increasing the coupling strength transitions the network from complete incoherence to coherent traveling waves via a chimera state.

Mutual correlation and chaotic behavior in phantom AdS black holes in both dynamic and static backgrounds

We research the mutual correlation and chaotic behavior in phantom Anti-de Sitter (AdS) black holes based mainly on shock wave geometry. In the static background of the phantom spacetime, by analyzing mutual correlation, we find out that the greater the phantom dark energy parameter is, the smaller the mutual correlation becomes, which implies phantom dark energy can destroy bilateral correlation of the two sub-regions from the AdS boundary. In the dynamic background, the results show that, the greater the applied energy perturbation is, the smaller the mutual correlation becomes. Especially, the bigger the phantom dark energy is exerted, the faster the mutual correlation destroys. This is similar to the butterfly effect of chaos theory.

Finite-time stability control with hardware-in-the-loop testing of a chaotic permanent magnet synchronous motor

Stability of Permanent Magnet Synchronous Motor (PMSM) is of prime importance for the successful operation of the drive system, which is found to be dependent on initial operating conditions showing chaotic characteristic. Therefore, the paper presents the stabilization of a chaotic PMSM system through the finite-time stability approach. From this aspect, three controllers have been developed which are effective for the suppression of machine chaotic behaviour. Proposed controllers are simpler and numerically stable in comparison with previous works. Furthermore, performance comparisons of the proposed controllers are also presented in term of their settling time and peak overshoot. Selection among the proposed controllers depends on the consideration of a particular operating index (settling time or peak overshoot) wherein, performance also varies on system parameter λ. Performance evaluation of proposed controllers has been presented in MATLAB environment. In order to validate the obtained MATLAB results, a real-time analysis has been carried out using Typhoon Hardware-in-the-loop (HIL) emulator. The comparison shows that both results follow the same trend and are closely coordinated with each other, which validates the proposed controllers.

Emergence of chaos in an electroactive artificial muscle PVC gel under state-varying electromechanical parameters

The deformation mechanism of artificial muscle materials based on the natural method has always been one of the frontier topics in the field of biophysics. Electroactive PVC gel is a composite material that mimics the response of natural muscles, making it highly advantageous for various bionic mechanical applications. The existing literature primarily focuses on enhancing the performance and exploring applications of PVC gel, with particular emphasis on optimizing its ability to simulate muscle deformation. However, the existing literature lacks research on the theoretical mechanism and modeling of large deformation, particularly regarding the investigation of nonlinear dynamic behavior under state-varying electromechanical parameters. The present research establishes a novel theoretical model for PVC gel and investigates its dynamic response to voltage, mechanical force, and inertia force. Firstly, the electromechanical constitutive model of PVC gel actuator was elucidated, and the nonlinear dynamics control equation of PVC gel actuator was established based on the Gent model. Secondly, we investigate the vibration characteristics of PVC gel material by analyzing four key parameters: size, tensile strength, applied voltage amplitude, and excitation frequency. Through this analysis, we determine the relevant parameters and operational range of the PVC gel actuator when subjected to voltage excitation. Meanwhile, the phase trajectories and Poincaré maps were generated based on the control equation calculation results to investigate the vibration stability and chaotic characteristics of the PVC gel actuator under electromechanical coupling conditions. The deformation of PVC gel is influenced by various nonlinear factors, wherein the alteration in frequency induces harmonic, subharmonic, and super-harmonic resonance phenomena within the frequency response. By revising the size, voltage, and frequency of these three factors, with particular emphasis on regulating the frequency, chaotic behavior in the PVC gel will be facilitated. Finally, the bifurcation analysis of the electromechanical vibration behavior of the PVC gel actuator was carried out, and the stability of the PVC gel vibration was further quantitatively determined by the Lyapunov exponent.

Bifurcation analysis, and exact solutions of the two-mode Cahn–Allen equation by a novel variable coefficient auxiliary equation method

This document elaborates on a newly introduced analytical method known as the “Variable Coefficient Generalized Abel Equation Method,” as proposed by Hashemi in Hashemi (2024), designed specifically for addressing the two-mode Cahn–Allen equation. Diverging from conventional techniques that heavily rely on constant coefficient ordinary differential equations and auxiliary ordinary differential equations, our method innovatively incorporates variable coefficient ordinary differential equations within a sub-equation framework. Demonstrating its versatility, we apply this innovative technique to the two-mode Cahn–Allen equation, showcasing its effectiveness and efficiency through the derivation of analytical solutions. Notably, this method emerges as a promising tool for tackling complex nonlinear partial differential equations prevalent in fluid dynamics and wave propagation scenarios. Beyond merely expanding the repertoire of available analytical tools, our approach contributes to advancing solutions for various models within the realm of mathematical physics. Various forms of exact solutions, including exponential-type solutions, Kink solitons, dark solitons, and bright soliton solutions, are obtained for the model under consideration. Moreover, we delve into the analysis of bifurcation, chaotic behavior, and sensitivity within the context of the two-mode Cahn–Allen model, further enhancing the depth and breadth of our study. Three equilibria are analyzed across various classifications, including center point, focus point, saddle point, and node point. Chaotic behavior of the corresponding dynamical system is considered by adding the function ω1sin(ω2ζ). Lastly, sensitivity analysis of the system is conducted by examining different parameters of the model and imposing noise to the initial conditions.

Quantifying Quantum Chaos through Microcanonical Distributions of Entanglement

A characteristic feature of “quantum chaotic” systems is that their eigenspectra and eigenstates display universal statistical properties described by random matrix theory (RMT). However, eigenstates of local systems also encode structure beyond RMT. To capture this feature, we introduce a framework that allows us to compare the properties of eigenstates in local systems with those of pure random states. In particular, our framework defines a notion of distance between quantum state ensembles that utilizes the Kullback-Leibler divergence to compare the microcanonical distribution of entanglement entropy (EE) of eigenstates with a reference RMT distribution generated by pure random states (with appropriate constraints). This notion gives rise to a quantitative metric for quantum chaos that not only accounts for averages of the distributions but also higher moments. The differences in moments are compared on a highly resolved scale set by the standard deviation of the RMT distribution, which is exponentially small in system size. As a result, the metric can distinguish between chaotic and integrable behaviors and, in addition, quantify and compare the of chaos (in terms of proximity to RMT behavior) between two systems that are assumed to be chaotic. We implement our framework in local, minimally structured, Floquet random circuits, as well as a canonical family of many-body Hamiltonians, the mixed-field Ising model (MFIM). Importantly, for Hamiltonian systems, we find that the reference random distribution must be appropriately constrained to incorporate the effect of energy conservation in order to describe the ensemble properties of midspectrum eigenstates. The metric captures deviations from RMT across all models and parameters, including those that have been previously identified as strongly chaotic, and for which other diagnostics of chaos such as level spacing statistics look strongly thermal. In Floquet circuits, the dominant source of deviations is the second moment of the distribution, and this persists for all system sizes. For the MFIM, we find significant variation of the KL divergence in parameter space. Notably, we find a small region where deviations from RMT are minimized, suggesting that “maximally chaotic” Hamiltonians may exist in fine-tuned pockets of parameter space. Published by the American Physical Society 2024

A fixed point analysis of fractional dynamics of heat transfer in chaotic fluid layers

This manuscript aims to analyze the chaotic fluid layer heating from below, using the Atangana–Baleanu derivative to characterize the existence and uniqueness of this chaotic model based on the provided fixed point data. First, with suitable examples, we demonstrate the existence and uniqueness of fixed point solutions within the framework of controlled metric spaces. Secondly, we study the chaotic behavior of water with various Rayleigh numbers by experimenting with it while keeping a constant Prandtl number. Furthermore, the role of the Rayleigh number in magnifying chaotic behaviors is examined, with graphical simulations demonstrating the effect of different fractional orders of the Atangana–Baleanu derivative. This research provides insights into the dynamics of chaotic systems and their practical applications in controlled situations.

Controlling chaos and codimension-two bifurcation in a discrete fractional-order Brusselator model

This paper explores the qualitative behavior of a discrete fractional–order Brusselator model. We analyze the local dynamics of the model around its fixed point and determine its topological classification. We perform the bifurcation analysis for both codimension-one and codimension-two cases to examine the system behavior near critical parameter values. Using normal form theory and center manifold theorem (CMT), we prove that the model exhibits period-doubling bifurcation around its interior fixed point. We also study the existence and direction of Neimark–Sacker bifurcation using normal form theory. For codimension-two bifurcation, we show that the model undergoes 1:2, 1:3, and 1:4 resonances by applying normal form theory and suitable affine transformations. The system displays a rich variety of bifurcations, including quasi–periodicity, periodic orbits, chaotic behavior, and resonance bifurcation. Furthermore, the existence of chaos is discussed in the sense of Marotto, and a novel chaos control method is proposed for discrete Brusselator model using an extended pole–placement approach. This modified approach is more suitable for codimension-two bifurcation situations. Numerical simulations are used to illustrate the theoretical discussion.

Chaos Control and Synchronization of a New Fractional Laser Chaotic System

In this article, we introduce a new fractional laser chaotic system derived from the Lorenz–Haken equations. We investigate the complex dynamics of the proposed system consisting chaos, stability, control and synchronization of chaos. Moreover, we numerically reveal the nonlinear dynamics of the fractional laser chaotic system through the phase portraits, time histories and bifurcation diagrams. Also, we indicate the chaotic behaviors of the system by means of Lyapunov exponents, bifurcation diagrams versus all parameters along the state variables, phase portraits and time histories with different trajectories and initial conditions. The necessary conditions to eliminate the chaotic vibration of the system are obtained via the feedback control procedure. Meanwhile, a synchronization mechanism based on the feedback control technique is achieved for coupled fractional laser chaotic systems. Furthermore, we show that the fractional derivative order is very effective on reducing the irregular and chaotic behaviors of the system.

Floquet analysis perspective of driven light–matter interaction models

In this paper, we analyze the harmonically driven Jaynes–Cummings and Lipkin–Meshkov–Glick models using both numerical integration of time-dependent Hamiltonians and Floquet theory. For a separation of time scales between the drive and intrinsic Rabi oscillations in the former model, the driving results in an effective periodic reversal of time. The corresponding Floquet Hamiltonian is a Wannier–Stark model, which can be analytically solved. Despite the chaotic nature of the driven Lipkin–Meshkov–Glick model, moderate system sizes can display qualitatively different behaviors under varying system parameters. Ergodicity arises in systems that are neither adiabatic nor diabatic, owing to repeated multi-level Landau–Zener transitions. Chaotic behavior, observed in slow driving, manifests as random jumps in the magnetization, suggesting potential utility as a random number generator. Furthermore, we discuss both models in terms of a Floquet Fock state lattice.

Comparative Analysis of the Chaotic Behavior of a Five-Dimensional Fractional Hyperchaotic System with Constant and Variable Order

Comparative Analysis of the Chaotic Behavior of a Five-Dimensional Fractional Hyperchaotic System with Constant and Variable Order

Design and Analysis of a Novel Fractional-Order System with Hidden Dynamics, Hyperchaotic Behavior and Multi-Scroll Attractors

The design of chaotic systems with complex dynamic behaviors has always been a key aspect of chaos theory in engineering applications. This study introduces a novel fractional-order system characterized by hidden dynamics, hyperchaotic behavior, and multi-scroll attractors. By employing fractional calculus, the system’s order is extended beyond integer values, providing a richer dynamic behavior. The system’s hidden dynamics are revealed through detailed numerical simulations and theoretical analysis, demonstrating complex attractors and bifurcations. The hyperchaotic nature of the system is verified through Lyapunov exponents and phase portraits, showing multiple positive exponents that indicate a higher degree of unpredictability and complexity. Additionally, the system’s multi-scroll attractors are analyzed, showcasing their potential for secure communication and encryption applications. The fractional-order approach enhances the system’s flexibility and adaptability, making it suitable for a wide range of practical uses, including secure data transmission, image encryption, and complex signal processing. Finally, based on the proposed fractional-order system, we designed a simple and efficient medical image encryption scheme and analyzed its security performance. Experimental results validate the theoretical findings, confirming the system’s robustness and effectiveness in generating complex chaotic behaviors.

Forecasting chaotic behavior of the drill bit in real-time using a DNN model integrated with rock surface morphology

The multiple regenerative effects during the drilling process cause the drilling system to exhibit complex motions, even leading to chaotic behavior. Additionally, due to the inability to obtain real-time rock parameters, traditional drilling dynamics models and time-series forecasting models cannot accurately forecast the real-time operational status of the drilling system. In this paper, a deep neural network (DNN) model integrated with rock surface morphology model is proposed to forecast the real-time operational status of the drill bit. This new model can permanently record the changes in rock surface morphology and multiple regenerative effects during the drilling process, enhancing the perception of rock parameters and improving the forecasting accuracy of the drill bit’s operational status. Experimental results show that the new model forecasts the drill bit’s operational status with higher accuracy compared to other typical time-series forecasting models without integrating surface morphology model. Furthermore, this new model demonstrates superior convergence speed, stability, and generalization ability compared to other models. In addition, the influence of hyperparameters on the forecast accuracy of the new model has also been studied. These results play an important role in forecasting and preventing adverse vibrations such as stick slip, impact, reversal, and rebound of the drill bit.

Chaotic Pattern and Solitary Solutions for the (21)-Dimensional Beta-Fractional Double-Chain DNA System

The dynamical behavior of the double-chain deoxyribonucleic acid (DNA) system holds significant implications for advancing the understanding of DNA transmission laws in the realms of biology and medicine. This study delves into the investigation of chaos patterns and solitary wave solutions for the (2+1) Beta-fractional double-chain DNA system, employing the theory of planar dynamical systems and the method of complete discrimination system for polynomials (CDSP). The results demonstrate a diverse spectrum of solitary wave solutions, sensitivity to perturbations, and manifestations of chaotic behavior within the system. Through the utilization of the complete discrimination system for polynomials, a multitude of novel solitary wave solutions, encompassing periodic, solitary wave, and Jacobian elliptic function solutions, were systematically constructed. The influence of Beta derivatives on the solutions was elucidated through parameter comparison analysis, emphasizing the innovative nature of this study. These findings underscore the potential of this system in unraveling various biologically significant DNA transmission mechanisms.

Sinusoidal control strategy applied to continuous stirred‐tank reactors: Asymptotic and exponential convergence

AbstractThe main goal of this proposal is to present a class of nonlinear controllers for regulation and set point changes in continuous chemical reactors. The proposed control law has in its mathematical structure a proportional term of the regulation error to provide closed‐loop stability and a sinusoidal term, which can compensate for the nonlinearities of the plant. The closed‐loop stability of the plant is demonstrated via Lyapunov analysis, which reveals an asymptotic convergence of the control output to the required set points. Furthermore, the analysis of the regulation error's dynamic under the considered assumptions leads us to conclude that exponential stability is also reached. The controller is implemented via numerical experiments in two examples to generalize the applicability of the proposed approach by considering continuous stirred‐tank reactors models. The first case considers autocatalytic chemical oscillatory reactions that induce chaotic behaviour. For the second case, a process of acetone, butanol, and ethanol (ABE) fermentation through Clostridium acetobutylicum is considered. The proposed strategy shows an adequate performance because it can reach the required set point without long time settings and overshoot. A comparison with a smooth sliding‐mode and a standard proportional‐integral (PI) controller indicates the advantages of the proposed control approach.

Revisiting Lorenz’s Error Growth Models: Insights and Applications

This entry examines Lorenz’s error growth models with quadratic and cubic hypotheses, highlighting their mathematical connections to the non-dissipative Lorenz 1963 model. The quadratic error growth model is the logistic ordinary differential equation (ODE) with a quadratic nonlinear term, while the cubic model is derived by replacing the quadratic term with a cubic one. A variable transformation shows that the cubic model can be converted to the same form as the logistic ODE. The relationship between the continuous logistic ODE and its discrete version, the logistic map, illustrates chaotic behaviors, demonstrating computational chaos with large time steps. A variant of the logistic ODE is proposed to show how finite predictability horizons can be determined, emphasizing the continuous dependence on initial conditions (CDIC) related to stable and unstable asymptotic values. This review also presents the mathematical relationship between the logistic ODE and the non-dissipative Lorenz 1963 model.

Exploring chaos and bifurcation in a discrete prey–predator based on coupled logistic map

This research paper investigates discrete predator-prey dynamics with two logistic maps. The study extensively examines various aspects of the system’s behavior. Firstly, it thoroughly investigates the existence and stability of fixed points within the system. We explores the emergence of transcritical bifurcations, period-doubling bifurcations, and Neimark-Sacker bifurcations that arise from coexisting positive fixed points. By employing central bifurcation theory and bifurcation theory techniques. Chaotic behavior is analyzed using Marotto’s approach. The OGY feedback control method is implemented to control chaos. Theoretical findings are validated through numerical simulations.

A nonlinear breathing crack model for characterization of chaotic dynamics of damaged large-amplitude vibrating plates

Large-amplitude vibrating plates (LVPs) inherently exhibit intricate chaotic behaviors stemming from geometric nonlinearity. Such chaotic dynamics could be substantially influenced by the occurrence of a breathing crack, characterized by a cyclic opening and closing process. However, current models for breathing cracks primarily focus on beam-like structures and often rely on the assumption of periodic dynamical responses, which is inadequate for depicting the chaotic vibrations experienced by cracked LVPs. To that end, this study presents a novel nonlinear breathing crack model (NBCM) to precisely characterize the breathing effect on chaotic dynamics of large-amplitude vibrating plates containing a parallel part-through breathing crack. By refining the line spring model and devising a mathematical expression for crack parameters and dynamical responses, the NBCM-based governing equation for cracked LVPs is derived with a Duffing type expression having a quadratic term. Through the utilization of the harmonic balance method and the Hilbert–Hughes–Taylor algorithm, both analytical and numerical dynamic responses of the cracked LVP based on the NBCM are obtained. Comprehensive nonlinearity analyses are then conducted, including hardening nonlinearity, breathing nonlinearity, and bifurcation nonlinearity. Comparison results demonstrate the superiority of the presented nonlinear breathing crack model in characterizing the chaotic dynamics of LVPs with breathing cracks. Accordingly, the presented NBCM shows significant potential for identifying abnormalities within practical engineering plate-like structures when considering the effects of crack breathing.